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Simplifying 8x * x + 13x + -23 = 0 Multiply x * x 8x2 + 13x + -23 = 0 Reorder the terms: -23 + 13x + 8x2 = 0 Solving -23 + 13x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.875 + 1.625x + x2 = 0 Move the constant term to the right: Add '2.875' to each side of the equation. -2.875 + 1.625x + 2.875 + x2 = 0 + 2.875 Reorder the terms: -2.875 + 2.875 + 1.625x + x2 = 0 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + 1.625x + x2 = 0 + 2.875 1.625x + x2 = 0 + 2.875 Combine like terms: 0 + 2.875 = 2.875 1.625x + x2 = 2.875 The x term is 1.625x. Take half its coefficient (0.8125). Square it (0.66015625) and add it to both sides. Add '0.66015625' to each side of the equation. 1.625x + 0.66015625 + x2 = 2.875 + 0.66015625 Reorder the terms: 0.66015625 + 1.625x + x2 = 2.875 + 0.66015625 Combine like terms: 2.875 + 0.66015625 = 3.53515625 0.66015625 + 1.625x + x2 = 3.53515625 Factor a perfect square on the left side: (x + 0.8125)(x + 0.8125) = 3.53515625 Calculate the square root of the right side: 1.88020112 Break this problem into two subproblems by setting (x + 0.8125) equal to 1.88020112 and -1.88020112.Subproblem 1
x + 0.8125 = 1.88020112 Simplifying x + 0.8125 = 1.88020112 Reorder the terms: 0.8125 + x = 1.88020112 Solving 0.8125 + x = 1.88020112 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + x = 1.88020112 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + x = 1.88020112 + -0.8125 x = 1.88020112 + -0.8125 Combine like terms: 1.88020112 + -0.8125 = 1.06770112 x = 1.06770112 Simplifying x = 1.06770112Subproblem 2
x + 0.8125 = -1.88020112 Simplifying x + 0.8125 = -1.88020112 Reorder the terms: 0.8125 + x = -1.88020112 Solving 0.8125 + x = -1.88020112 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + x = -1.88020112 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + x = -1.88020112 + -0.8125 x = -1.88020112 + -0.8125 Combine like terms: -1.88020112 + -0.8125 = -2.69270112 x = -2.69270112 Simplifying x = -2.69270112Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.06770112, -2.69270112}
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